Factorising Quadratics Quickly

Factorising Quadratics: A Core IGCSE Skill

Factorising quadratic expressions is one of the most frequently tested skills in IGCSE Maths. It appears in algebraic manipulation questions, in solving quadratic equations, in simplifying algebraic fractions, and even in some coordinate geometry problems. If you can factorise quickly and accurately, you have a massive advantage on exam day.

This guide covers three levels of factorising difficulty, from the basics through to the methods needed for the hardest Paper 4 questions.

Level 1: Simple Quadratics (a = 1)

When the quadratic has the form x² + bx + c (with a coefficient of 1 for x²), factorising is straightforward.

You need two numbers that:

Multiply to give c (the constant term)
Add to give b (the coefficient of x)

Example: Factorise x² + 7x + 12

What multiplies to 12? The pairs are: 1 and 12, 2 and 6, 3 and 4
Which pair adds to 7? Answer: 3 and 4
Result: (x + 3)(x + 4)

Example: Factorise x² − 2x − 15

What multiplies to −15? Think about factor pairs where one is negative: (−5, 3), (5, −3), (−15, 1), (15, −1)
Which pair adds to −2? Answer: −5 and 3
Result: (x − 5)(x + 3)

The speed tip here is to start with the constant term’s factors and mentally check each pair against the coefficient of x. With practice, this becomes almost instantaneous.

Handling Signs

The signs in the quadratic tell you about the signs of the two numbers:

x² + bx + c (both positive): both numbers are positive
x² − bx + c (negative middle, positive end): both numbers are negative
x² + bx − c (positive middle, negative end): one positive (larger), one negative
x² − bx − c (both negative parts): one negative (larger magnitude), one positive

Knowing these patterns lets you narrow down the possible pairs immediately, saving valuable seconds.

Level 2: Taking Out a Common Factor First

Sometimes a quadratic has a common factor across all terms. Always take it out first.

Example: Factorise 3x² + 12x + 9

Take out the common factor of 3: 3(x² + 4x + 3)
Now factorise the simple quadratic: 3(x + 1)(x + 3)

Example: Factorise 2x² − 8x

Common factor of 2x: 2x(x − 4)

Missing the common factor makes the rest of the factorising much harder. Always check for one before attempting any other method.

Level 3: Harder Quadratics (a > 1)

When the coefficient of x² is not 1, such as 2x² + 7x + 3, you need a more systematic approach. There are two popular methods.

Method 1: The AC Method (Grouping)

For ax² + bx + c:

Multiply a and c: 2 × 3 = 6
Find two numbers that multiply to ac and add to b: multiply to 6 and add to 7. That is 1 and 6
Rewrite the middle term using these numbers: 2x² + x + 6x + 3
Group in pairs: x(2x + 1) + 3(2x + 1)
Factor out the common bracket: (x + 3)(2x + 1)

Method 2: Trial and Improvement

Write out the bracket structure and try combinations:

The x² term is 2x², so the brackets start with (2x …)(x …)
The constant is 3, so the possible endings are (… 1)(… 3) or (… 3)(… 1)
Try (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. That works.

Method 2 can be faster with practice but is less systematic. Method 1 always works and is more reliable under exam pressure.

The Difference of Two Squares

This special case deserves emphasis because it appears so often:

a² − b² = (a + b)(a − b)

Examples:

x² − 25 = (x + 5)(x − 5)
9x² − 16 = (3x + 4)(3x − 4)
4y² − 1 = (2y + 1)(2y − 1)

Recognition is instant once you train yourself to spot two perfect squares separated by a minus sign. There is no middle term.

Factorising to Solve Equations

The main reason you factorise quadratics in IGCSE Maths is to solve quadratic equations. Once you have factorised, set each bracket equal to zero.

Example: Solve x² − 3x − 10 = 0

Factorise: (x − 5)(x + 2) = 0
Set each bracket to zero: x − 5 = 0 or x + 2 = 0
Solutions: x = 5 or x = −2

Remember: a quadratic equation usually has two solutions. If you only give one, you will lose a mark.

When Factorising Does Not Work

Not every quadratic factorises neatly with integers. When it does not, you have two backup methods:

The Quadratic Formula: x = (−b ± √(b² − 4ac)) / 2a

This always works. It is given on the formula sheet for IGCSE, so you do not need to memorise it, but you do need to be able to use it quickly and accurately.

Completing the Square: rewrite ax² + bx + c in the form a(x + p)² + q

This method is also used to find the vertex of a parabola. It is covered in detail in a separate article.

Speed Drills for Exam Preparation

To build factorising speed:

Set a timer for 5 minutes and factorise as many simple quadratics as you can
Gradually introduce harder quadratics with a > 1
Practice recognising the difference of two squares on sight
Mix in questions where you need to take out a common factor first
Always check your answer by expanding the brackets to verify

The goal is for simple factorising to take less than 30 seconds and harder factorising to take less than a minute. This leaves you more time for the challenging parts of the exam.

Common Errors to Watch For

Forgetting to check for a common factor first
Getting the signs wrong, especially when c is negative
Not giving both solutions when solving an equation
Confusing factorising with expanding — they are opposite operations
Making arithmetic errors in the AC method — always verify by expanding your answer

Where Factorising Appears in the Exam

Keep an eye out for factorising opportunities in these question types:

Direct “factorise” questions
Solving quadratic equations
Simplifying algebraic fractions
Finding x-intercepts of a parabola
Solving simultaneous equations where one equation is quadratic

Factorising is a foundational skill that supports many other parts of the IGCSE syllabus. Time invested in mastering it pays dividends across the entire exam.

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